Fast Fourier Transform

If f is an integrable function of a variable \(t\in (-\ ,\,\infty )\), then its Fourier integral or Fourier transform may be defined as the complex function F of a variable \(\omega \in (-\infty ,\,\infty )\) where \(F(\omega ):=\int _{-\infty }^{\infty }f(t)\,e^{-i\omega t}\, dt,\quad i=\sqrt{-1}\). The most important fact about definition (1) is that if it is viewed as an integral equation, its solution under very general conditions is Equation (2) is called the inverse transform of the direct transform of (1), and the two constitute the Fourier transform pair.